A000667 -id:A000667 - cp's OEIS frontend

A296793 a(n) = n! * [x^n] exp(x)*(sec(x) + tan(x))^n.

1, 2, 9, 67, 705, 9601, 160429, 3175579, 72638209, 1884974185, 54709142101, 1755923320559, 61748847320545, 2360991253910069, 97518218630249005, 4327060674324941491, 205272207854416078849, 10367500700785078039473, 555414837143457708584101, 31458118283019682610004279

Offset: 0

Ilya Gutkovskiy, Dec 20 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..300
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps)
N. J. A. Sloane, Transforms.
Index entries for sequences related to boustrophedon transform

Cf. A000111, A000667, A292756, A292759, A292976.

Main diagonal of A322268.

Table[n! SeriesCoefficient[Exp[x] (Sec[x] + Tan[x])^n, {x, 0, n}], {n, 0, 19}]

a(n) = Vec(serlaplace(exp(x)*(1/cos(x) + tan(x))^n))[n+1] \\ Iain Fox, Dec 20 2017

a(n) ~ c * d^n * n^n, where d = 1.12712316036287986633533456353714856005183790513784733... and c = 1.61865092826915643845148401952113086265743345... - Vaclav Kotesovec, Dec 21 2017

A317022 Expansion of e.g.f. sec(exp(x) - 1) + tan(exp(x) - 1).

Original entry on oeis.org

1, 1, 2, 6, 25, 132, 838, 6209, 52592, 501238, 5308295, 61839954, 785915626, 10820482467, 160436371306, 2548722840218, 43188812459297, 777586865332600, 14823480294719570, 298285781617278681, 6318170247815155180, 140520406400556170514, 3274091838364580459623

Offset: 0

Ilya Gutkovskiy, Jul 19 2018

nonn

Stirling transform of A000111.

Alois P. Heinz, Table of n, a(n) for n = 0..445
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000111, A000667, A080832, A139134.

b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
a:= n-> add(b(j, 0)*Stirling2(n, j), j=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 19 2018

nmax = 22; CoefficientList[Series[Sec[Exp[x] - 1] + Tan[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
e[n_] := e[n] = (2 I)^n If[EvenQ[n], EulerE[n, 1/2], EulerE[n, 0] I]; a[n_] := a[n] = Sum[StirlingS2[n, k] e[k], {k, 0, n}]; Table[a[n], {n, 0, 22}]

from itertools import accumulate
from sympy.functions.combinatorial.numbers import stirling
def A317022(n): # generator of terms
    if n == 0: return 1
    blist, c = (0,1), 0
    for k in range(1,n+1):
        c += stirling(n,k)*blist[-1]
        blist = tuple(accumulate(reversed(blist),initial=0))
    return c # Chai Wah Wu, Apr 18 2023

a(n) = Sum_{k=0..n} Stirling2(n,k)*A000111(k).

a(n) ~ n! * 4 / ((2+Pi) * (log(1+Pi/2))^(n+1)). - Vaclav Kotesovec, Sep 25 2019

A348580 Expansion of e.g.f. exp(x) / (1 - sin(x)).

Original entry on oeis.org

1, 2, 5, 15, 53, 217, 1015, 5355, 31513, 204857, 1458875, 11299695, 94600373, 851419597, 8198959735, 84124450035, 916270051633, 10559066809937, 128362804540595, 1641730799916375, 22037407161945293, 309782122281453877, 4551072446448773455, 69747642031977698715

Offset: 0

Ilya Gutkovskiy, Oct 24 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..484

Cf. A000111, A000667, A186364, A330046, A348587.

b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
a:= n-> add(binomial(n, k)*b(k+1, 0), k=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 24 2021

nmax = 23; CoefficientList[Series[Exp[x]/(1 - Sin[x]), {x, 0, nmax}], x] Range[0, nmax]!

my(x='x+O('x^40)); Vec(serlaplace(exp(x)/(1-sin(x)))) \\ Michel Marcus, Oct 24 2021

a(n) = Sum_{k=0..n} binomial(n,k) * A000111(k+1).

a(n) ~ 2^(n + 7/2) * n^(n + 3/2) / (Pi^(n + 3/2) * exp(n - Pi/2)). - Vaclav Kotesovec, Oct 25 2021

A059235 The array in A059219 read by antidiagonals in the direction in which it was constructed.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 3, 5, 0, 5, 8, 12, 15, 0, 15, 27, 39, 48, 55, 0, 55, 103, 152, 190, 221, 239, 0, 239, 460, 680, 871, 1025, 1137, 1199, 0, 1199, 2336, 3471, 4493, 5374, 6062, 6553, 6810, 0, 6810, 13363, 19903, 25958, 31351, 35884, 39399, 41847, 43108, 0

Offset: 1

N. J. A. Sloane, Jan 18 2001

			The array begins
1 1 0 5 0 55 0 ...
0 1 3 5 48 55 ...
2 2 8 39 103 ...
0 12 27 152 ...
15 15 190 ...
0 221 ...

Cf. A000667, A059216, A059219, A059217, A059220.

max = 10; t[0, 0] = 1; t[0, ?EvenQ] = 0; t[?OddQ, 0] = 0; t[n_, k_] /; OddQ[n+k](*up*):= t[n, k] = t[n+1, k-1] + Sum[t[n, j], {j, 0, k-1}]; t[n_, k_] /; EvenQ[n+k](*down*):= t[n, k] = t[n-1, k+1] + Sum[t[j, k], {j, 0, n-1}]; Table[tn = Table[t[n-k, k], {k, 0, n}]; If[OddQ[n], tn, tn // Reverse] , {n, 0, max}] // Flatten (* Jean-François Alcover, Nov 20 2012 *)

More terms from Larry Reeves (larryr(AT)acm.org), Jan 24 2001

A059505 Transform of A059502 applied to sequence 2,3,4,...

Original entry on oeis.org

2, 5, 14, 40, 114, 323, 910, 2551, 7120, 19796, 54852, 151525, 417434, 1147145, 3145394, 8606848, 23507190, 64093031, 174474790, 474261691, 1287398452, 3490267820, 9451319304, 25565098825, 69080289074

Offset: 1

Floor van Lamoen, Jan 19 2001

The second row of the array A059503.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for sequences related to boustrophedon transform
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Cf. A000667, A059216, A059219, A059502.

LinearRecurrence[{6,-11,6,-1},{2,5,14,40}, 50] (* or *) Rest[CoefficientList[Series[x*(2 - 7*x + 6*x^2 - x^3)/(1 - 3*x + x^2)^2, {x,0,50}], x]] (* G. C. Greubel, Sep 10 2017 *)

x='x+O('x^50); Vec(x*(2-7*x+6*x^2-x^3)/(1-3*x+x^2)^2) \\ G. C. Greubel, Sep 10 2017

G.f.: x*(2 - 7*x + 6*x^2 - x^3)/(1 - 3*x + x^2)^2.
From G. C. Greubel, Sep 10 2017: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
a(n) = ((3 - n)*Fibonacci(2*n) + (5 + 3*n)*Fibonacci(2*n - 1))/5. (End)

A059506 Transform of A059502 applied to sequence 3,4,5,...

Original entry on oeis.org

3, 7, 19, 53, 148, 412, 1143, 3161, 8717, 23977, 65798, 180182, 492459, 1343563, 3659623, 9953117, 27031768, 73320496, 198632607, 537507677, 1452978593, 3923762257, 10586222474, 28536313898, 76859031123

Offset: 1

Floor van Lamoen, Jan 19 2001

The third row of the array A059503.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for sequences related to boustrophedon transform
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Cf. A000667, A059216, A059219, A059502.

LinearRecurrence[{6,-11,6,-1},{3,7,19,53},30] (* Harvey P. Dale, Jul 30 2015 *)
Rest[CoefficientList[Series[x*(1 - x)*(2*x^2 - 8*x + 3)/(x^2 - 3*x + 1)^2, {x,0,50}], x]] (* G. C. Greubel, Sep 10 2017 *)

Vec(x*(1-x)*(2*x^2-8*x+3)/(x^2-3*x+1)^2 + O(x^30)) \\ Michel Marcus, Sep 09 2017

From Colin Barker, Nov 30 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
G.f.: x*(1-x)*(2*x^2-8*x+3)/(x^2-3*x+1)^2. (End)
a(n) = ((3 - n)*Fibonacci(2*n) + (10 + 3*n)*Fibonacci(2*n - 1))/5. - G. C. Greubel, Sep 10 2017

A059507 Transform of A059502 applied to sequence 4,5,6,...

Original entry on oeis.org

4, 9, 24, 66, 182, 501, 1376, 3771, 10314, 28158, 76744, 208839, 567484, 1539981, 4173852, 11299386, 30556346, 82547961, 222790424, 600753663, 1618558734, 4357256694, 11721125644, 31507528971, 84637773172

Offset: 1

Floor van Lamoen, Jan 19 2001

The fourth row of the array A059503.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for sequences related to boustrophedon transform
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Cf. A000667, A059216, A059219, A059502.

Rest[CoefficientList[Series[x*(1 - x)*(3*x^2 - 11*x + 4)/(x^2 - 3*x + 1)^2, {x, 0, 50}], x]] (* G. C. Greubel, Sep 10 2017 *)

Vec(x*(1-x)*(3*x^2-11*x+4)/(x^2-3*x+1)^2 + O(x^40)) \\ Michel Marcus, Sep 09 2017

From Colin Barker, Nov 30 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
G.f.: x*(1-x)*(3*x^2-11*x+4)/(x^2-3*x+1)^2. (End)
a(n) = ((3 - n)*Fibonacci(2*n) + (15 + 3*n)*Fibonacci(2*n - 1))/5. - G. C. Greubel, Sep 10 2017

A059508 Transform of A059502 applied to sequence 5,6,7,...

Original entry on oeis.org

5, 11, 29, 79, 216, 590, 1609, 4381, 11911, 32339, 87690, 237496, 642509, 1736399, 4688081, 12645655, 34080924, 91775426, 246948241, 663999649, 1784138875, 4790751131, 12856028814, 34478744044, 92416515221

Offset: 1

Floor van Lamoen, Jan 19 2001

The fifth row of the array A059503.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for sequences related to boustrophedon transform
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Cf. A000667, A059216, A059219, A059502, A059503.

Rest[CoefficientList[Series[x*(1-x)*(4*x^2 - 14*x + 5)/(x^2 - 3*x + 1)^2, {x, 0, 50}], x]] (* G. C. Greubel, Sep 10 2017 *)

Vec(-x*(x-1)*(4*x^2-14*x+5)/(x^2-3*x+1)^2 + O(x^40)) \\ Michel Marcus, Sep 09 2017

From Colin Barker, Nov 30 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
G.f.: x*(1-x)*(4*x^2-14*x+5)/(x^2-3*x+1)^2. (End)
a(n) = ((3 - n)*Fibonacci(2*n) + (20 + 3*n)*Fibonacci(2*n - 1))/5. - G. C. Greubel, Sep 10 2017

A059575 The array described in A059513 read by antidiagonals in the direction of construction.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 13, 19, 23, 1, 48, 87, 107, 116, 1, 243, 458, 635, 708, 736, 1, 1517, 2967, 4239, 5163, 5533, 5659, 1, 11562, 22824, 33291, 41772, 47733, 50031, 50796, 1, 103125, 204598, 301161, 385422, 452016, 497789, 515254, 521040

Offset: 1

Floor van Lamoen, Jan 23 2001

Cf. A000667, A035002, A059216, A059219, A059574.

Sequence contained two errors corrected by N. J. A. Sloane, Jun 14 2005

A292759 Expansion of e.g.f. exp(x)*(tan x + sec x)^3.

Original entry on oeis.org

1, 4, 16, 67, 304, 1519, 8386, 51007, 340024, 2469859, 19438606, 164899447, 1500636844, 14587478299, 150891959026, 1655133221887, 19192311085264, 234597922978339, 3015167371458646, 40651421300224327, 573707768015267284, 8458761578948943979, 130059537979390701466

Offset: 0

N. J. A. Sloane, Sep 26 2017

nonn

C. K. Cook, M. R. Bacon, and R. A. Hillman, Higher-order Boustrophedon transforms for certain well-known sequences, Fib. Q., 55(3) (2017), 201-208.

Cf. A000667, A292756.

nmax = 20; CoefficientList[Series[Exp[x]*(Tan[x]+Sec[x])^3, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 02 2019 *)

a(n) ~ 2^(n + 11/2) * n^(n + 5/2) / (Pi^(n + 5/2) * exp(n - Pi/2)). - Vaclav Kotesovec, Jun 02 2019

cp's OEIS Frontend

A296793 a(n) = n! * [x^n] exp(x)*(sec(x) + tan(x))^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A317022 Expansion of e.g.f. sec(exp(x) - 1) + tan(exp(x) - 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A348580 Expansion of e.g.f. exp(x) / (1 - sin(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A059235 The array in A059219 read by antidiagonals in the direction in which it was constructed.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A059505 Transform of A059502 applied to sequence 2,3,4,...

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A059506 Transform of A059502 applied to sequence 3,4,5,...

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A059507 Transform of A059502 applied to sequence 4,5,6,...

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A059508 Transform of A059502 applied to sequence 5,6,7,...

Views

Author